Trigonometric equations

IB Mathematics: Analysis and Approaches HL· Topic 3: Geometry and Trigonometry· 15 min read

1. Linear Trigonometric Equations★★☆☆☆⏱ 5 min

A linear trigonometric equation has the form $a sin (b x + c) = k$ , $a cos (b x + c) = k$ , or $a tan (b x + c) = k$ , where $a, b, c, k$ are constants. We first isolate the trigonometric term, then find solutions using the unit circle.

📘 Definition

Linear trigonometric equation

An equation where the trigonometric function is raised only to the first power, with no products of different trigonometric terms.

📐 Worked Example

Solve $2 sin x = 1$ for $0 \leq x < 2 π$

1
First, isolate $sin x$ :
2
$\sin x = \frac{1}{2}$
3
Find the principal solution in the first quadrant: $x = \frac{π}{6}$ . Sine is positive in both the first and second quadrants, so the second solution in $[0, 2 π)$ is:
4
$x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$
5
Both solutions fall within the given domain, so the final solutions are:
6
$x = \frac{\pi}{6}, \quad x = \frac{5\pi}{6}$

2. Quadratic Trigonometric Equations★★★☆☆⏱ 6 min

Quadratic trigonometric equations can be rearranged into the form $a u^{2} + b u + c = 0$ , where $u$ is a trigonometric function like $sin x$ , $cos x$ , or $tan x$ . We solve using factoring or the quadratic formula, then solve each resulting linear trigonometric equation.

📐 Worked Example

Solve $2 cos^{2} x - 3 cos x + 1 = 0$ for $0 \leq x \leq π$

1
Substitute $u = cos x$ to get a standard quadratic:
2
$2u^2 - 3u + 1 = 0$
3
Factor the quadratic:
4
$(2u - 1)(u - 1) = 0$
5
This gives two cases: $u = 1$ or $u = \frac{1}{2}$ . Solve each case for $x$ :
6
Case 1: $cos x = 1 ⟹ x = 0$ , which is in the domain. Case 2: $cos x = \frac{1}{2} ⟹ x = \frac{π}{3}$ (the other solution $\frac{5 π}{3}$ is greater than $π$ , so excluded).
7
Final solutions:
8
$x = 0, \quad x = \frac{\pi}{3}$

3. Solving Equations Using Identities★★★★☆⏱ 7 min

Many trigonometric equations require use of double-angle, compound-angle, or Pythagorean identities to simplify them into a solvable linear or quadratic form. Always look for opportunities to rewrite multiple angles or different trigonometric terms in terms of a single variable.

📐 Worked Example

Solve $sin 2 x = sin x$ for $0 \leq x < 2 π$

1
Use the double-angle identity $sin 2 x = 2 sin x cos x$ :
2
$2\sin x \cos x = \sin x$
3
Rearrange and factor out $sin x$ (do not divide both sides by $sin x$ , this will lose solutions):
4
$2\sin x \cos x - \sin x = 0 \implies \sin x (2\cos x - 1) = 0$
5
Set each factor equal to zero and solve: Case 1: $sin x = 0 ⟹ x = 0, π$ . Case 2: $2 cos x - 1 = 0 ⟹ cos x = \frac{1}{2} ⟹ x = \frac{π}{3}, \frac{5 π}{3}$ .
6
All solutions are within the domain, so the full set of solutions is:
7
$x = 0, \frac{\pi}{3}, \pi, \frac{5\pi}{3}$

4. General Solutions for Unrestricted Domains★★★★☆⏱ 5 min

When no domain is specified, we need to write the general solution that accounts for the periodicity of trigonometric functions. Sine and cosine have period $2 π$ , so we add $2 π n$ to all base solutions. Tangent has period $π$ , so we add $π n$ to base solutions, where $n \in Z$ .

📐 Worked Example

Find the general solution of $tan (x - \frac{π}{4}) = 1$

1
Let $θ = x - \frac{π}{4}$ . The general solution of $tan θ = 1$ is:
2
$\theta = \frac{\pi}{4} + \pi n, \quad n \in \mathbb{Z}$
3
Substitute back $θ = x - \frac{π}{4}$ :
4
$x - \frac{\pi}{4} = \frac{\pi}{4} + \pi n$
5
Rearrange to get the general solution:
6
$x = \frac{\pi}{2} + \pi n, \quad n \in \mathbb{Z}$

5. Common Pitfalls

Wrong move:

Only taking the principal calculator solution and ignoring solutions in other quadrants

Why:

Calculators only output one principal solution, but trigonometric equations have multiple solutions per period

Correct move:

Use the unit circle and the sign of the trigonometric ratio to find all base solutions in one period before adding period multiples

Wrong move:

Dividing both sides of an equation by a common trigonometric term (e.g. dividing $sin 2 x = sin x$ by $sin x$ )

Why:

This removes the case where the trigonometric term equals zero, losing all corresponding solutions

Correct move:

Rearrange the equation to factor out the common trigonometric term, keeping all possible solution cases

Wrong move:

Adding $2 π n$ as the period for tangent equations

Why:

Tangent repeats every $π$ , not $2 π$ , so this misses half of all general solutions

Correct move:

Always add $π n$ for tangent equations, and $2 π n$ for sine and cosine equations

Wrong move:

Failing to filter solutions to match the stated domain

Why:

After generating general solutions, it is easy to include solutions that fall outside the given interval

Correct move:

List all candidate solutions, then explicitly check each one to confirm it falls within the domain bounds before writing your final answer

Wrong move:

Leaving extraneous solutions untested after squaring both sides of an equation

Why:

Squaring both sides can introduce solutions that do not satisfy the original equation

Correct move:

Test every solution in the original equation to remove any invalid extraneous solutions

6. Quick Reference Cheatsheet

Equation Type	Key Method	General Solution Form
$sin θ = k$	Find base solutions via symmetry	$θ = α + 2 π n, θ = π - α + 2 π n, n \in Z$
$cos θ = k$	Find base solutions via symmetry	$θ = α + 2 π n, θ = - α + 2 π n, n \in Z$
$tan θ = k$	One base solution per period	$θ = α + π n, n \in Z$
Quadratic trig	Substitute $u = sin x / cos x / tan x$	Factor/solve quadratic, solve each linear case
Equation with multiple angles	Use angle identities to simplify	Adjust period for the multiple angle coefficient

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Solve quadratic trigonometric equation
2022 · 2
Find general solution of linear equation
2023 · 1
Solve with double angle identity

Going deeper

syllabusIB AA HL Topic 3 SyllabusOfficial syllabus reference

What's Next

Solving trigonometric equations is a core foundational skill for almost all remaining topics in IB AA HL. It is used regularly when integrating trigonometric functions, solving differential equations, analysing periodic models, and working with polar forms of complex numbers. Mastery of this topic will make working with these more advanced concepts significantly easier, as you will already be comfortable rearranging and solving trigonometric expressions. Next, you can deepen your understanding of trigonometric applications or extend your knowledge to related topics.